Question: Jeremiah makes $25\%$ of the three-point shots he attempts. For a warm up, Jeremiah likes to shoot three-point shots until he successfully makes one. Let $M$ be the number of shots it takes Jeremiah to successfully make his first three-point shot. Assume that the results of each shot are independent. Find the probability that Jeremiah's first successful shot occurs on his $3^{\text{rd}}$ attempt. You may round your answer to the nearest hundredth. $P(M=3)=$
Without a fancy calculator On each shot: $P({\text{success}})=0.25$ $P(\text{miss}})=0.75$ If his first successful shot occurs on his $3^{\text{rd}}$ attempt, then his sequence of results needs to be "miss, miss, success." $\begin{aligned} P(M=3)&=P(\text{miss}}, \text{miss}}, {\text{success}}) \\\\ &=(0.75})(0.75})({0.25}) \\\\ &=(0.75)^2(0.25) \\\\ &=0.140625 \end{aligned}$ $P(M=3)=0.140625$